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Theorem neindisj 22484
Description: Any neighborhood of an element in the closure of a subset intersects the subset. Part of proof of Theorem 6.6 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)
Hypothesis
Ref Expression
neips.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
neindisj (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑃 ∈ ((clsβ€˜π½)β€˜π‘†) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃}))) β†’ (𝑁 ∩ 𝑆) β‰  βˆ…)

Proof of Theorem neindisj
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 neips.1 . . . . . . . 8 𝑋 = βˆͺ 𝐽
21clsss3 22426 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋)
32sseld 3948 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑃 ∈ ((clsβ€˜π½)β€˜π‘†) β†’ 𝑃 ∈ 𝑋))
43impr 456 . . . . 5 ((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) β†’ 𝑃 ∈ 𝑋)
51isneip 22472 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃}) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑃 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
64, 5syldan 592 . . . 4 ((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃}) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑃 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
7 3anass 1096 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†)) ↔ (𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))))
81clsndisj 22442 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†)) ∧ (𝑔 ∈ 𝐽 ∧ 𝑃 ∈ 𝑔)) β†’ (𝑔 ∩ 𝑆) β‰  βˆ…)
97, 8sylanbr 583 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) ∧ (𝑔 ∈ 𝐽 ∧ 𝑃 ∈ 𝑔)) β†’ (𝑔 ∩ 𝑆) β‰  βˆ…)
109anassrs 469 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) ∧ 𝑔 ∈ 𝐽) ∧ 𝑃 ∈ 𝑔) β†’ (𝑔 ∩ 𝑆) β‰  βˆ…)
1110adantllr 718 . . . . . . . 8 (((((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) ∧ 𝑁 βŠ† 𝑋) ∧ 𝑔 ∈ 𝐽) ∧ 𝑃 ∈ 𝑔) β†’ (𝑔 ∩ 𝑆) β‰  βˆ…)
1211adantrr 716 . . . . . . 7 (((((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) ∧ 𝑁 βŠ† 𝑋) ∧ 𝑔 ∈ 𝐽) ∧ (𝑃 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ (𝑔 ∩ 𝑆) β‰  βˆ…)
13 ssdisj 4424 . . . . . . . . . 10 ((𝑔 βŠ† 𝑁 ∧ (𝑁 ∩ 𝑆) = βˆ…) β†’ (𝑔 ∩ 𝑆) = βˆ…)
1413ex 414 . . . . . . . . 9 (𝑔 βŠ† 𝑁 β†’ ((𝑁 ∩ 𝑆) = βˆ… β†’ (𝑔 ∩ 𝑆) = βˆ…))
1514necon3d 2965 . . . . . . . 8 (𝑔 βŠ† 𝑁 β†’ ((𝑔 ∩ 𝑆) β‰  βˆ… β†’ (𝑁 ∩ 𝑆) β‰  βˆ…))
1615ad2antll 728 . . . . . . 7 (((((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) ∧ 𝑁 βŠ† 𝑋) ∧ 𝑔 ∈ 𝐽) ∧ (𝑃 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ ((𝑔 ∩ 𝑆) β‰  βˆ… β†’ (𝑁 ∩ 𝑆) β‰  βˆ…))
1712, 16mpd 15 . . . . . 6 (((((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) ∧ 𝑁 βŠ† 𝑋) ∧ 𝑔 ∈ 𝐽) ∧ (𝑃 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ (𝑁 ∩ 𝑆) β‰  βˆ…)
1817rexlimdva2 3155 . . . . 5 (((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) ∧ 𝑁 βŠ† 𝑋) β†’ (βˆƒπ‘” ∈ 𝐽 (𝑃 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ (𝑁 ∩ 𝑆) β‰  βˆ…))
1918expimpd 455 . . . 4 ((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) β†’ ((𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑃 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ (𝑁 ∩ 𝑆) β‰  βˆ…))
206, 19sylbid 239 . . 3 ((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃}) β†’ (𝑁 ∩ 𝑆) β‰  βˆ…))
2120exp32 422 . 2 (𝐽 ∈ Top β†’ (𝑆 βŠ† 𝑋 β†’ (𝑃 ∈ ((clsβ€˜π½)β€˜π‘†) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃}) β†’ (𝑁 ∩ 𝑆) β‰  βˆ…))))
2221imp43 429 1 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑃 ∈ ((clsβ€˜π½)β€˜π‘†) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃}))) β†’ (𝑁 ∩ 𝑆) β‰  βˆ…)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆƒwrex 3074   ∩ cin 3914   βŠ† wss 3915  βˆ…c0 4287  {csn 4591  βˆͺ cuni 4870  β€˜cfv 6501  Topctop 22258  clsccl 22385  neicnei 22464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-top 22259  df-cld 22386  df-ntr 22387  df-cls 22388  df-nei 22465
This theorem is referenced by:  clslp  22515  flimclslem  23351  utop3cls  23619
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